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Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote the $\infty$-category of small stable $\infty$-categories and exact functors by $Cat^{\rm ex}_{\infty}$. Suppose I have given a functor $$ F \colon \mathcal{I} \times \mathcal{I}^{\rm op} \to Cat^{\rm ex}_{\infty} $$ where I am secretly regarding the category on the left as an $\infty$-category.

Are coends defined for $\infty$-functors as above? Does the coend of $F$ as above always exist?

If the answer to the above questions is "yes": Can we describe the weak Kan complex underlying the coend in some way?

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  • $\begingroup$ Interesting question. I was led to something similar when I asked how are weighted limits defined in this setting. But why are you limiting to stable $\infty$-categories? $\endgroup$
    – fosco
    Commented Mar 4, 2014 at 20:05

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There is some discussion of coends in $\infty$-categories in my PhD thesis (Sheffield, 2010). I think that, given that the target category $\mathrm{Cat}_\infty^\mathrm{ex}$ is cocomplete, this gives you one way to do it.

I have another way, which is arguably rather less fussy, but it is currently not written up in a presentable fashion.

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